= Iteration of a one-fixed-point Möbius transformation
{title2=$T^nz\longrightarrow z_0$}
A nonidentity <Möbius transformation> with only one <fixed point> is conjugate to a nonzero translation of the <Riemann sphere>. Its translation iterates are $z+nb$, tending to infinity in the topology of the <Riemann sphere> for finite $z$, with infinity itself fixed. Conjugating back proves that every point tends to the unique <fixed point>. This is convergence on the <sphere>; if the <fixed point> is finite, intermediate iterates can still pass through infinity.
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